Torsion pairs and Ringel duality for Schur algebras

TL;DR

This paper explores the structure of Schur algebras using torsion pairs, demonstrating Morita equivalences to their Ringel duals under specific conditions, advancing understanding of their algebraic properties.

Contribution

It introduces a functorial method involving torsion pairs to embed endomorphism algebras and establishes Morita equivalences for certain blocks of Schur algebras with their Ringel duals.

Findings

Blocks of classical and quantum Schur algebras with 2p^k simple modules are Morita equivalent to their Ringel duals.

The functorial approach provides new embeddings of endomorphism algebras of projective modules.

The results apply to both classical and quantum cases, broadening their algebraic understanding.

Abstract

Let $A$ be a finite-dimensional algebra over a field of characteristic $p>0$. We use a functorial approach involving torsion pairs to construct embeddings of endomorphism algebras of basic projective $A$--modules $P$ into those of the torsion submodules of $P$. As an application, we show that blocks of both the classical and quantum Schur algebras $S(2,r)$ and $S_q(2,r)$ are Morita equivalent as quasi-hereditary algebras to their Ringel duals if they contain $2p^k$ simple modules for some $k$.